PCh7 Lec 1 2

Chapter 7, Lectures 1 and 2 (previously delivered)

Quantum mechanics seeks to address all modes of motion/energy. As discussed at the beginning of Chapter 7, there are three types of motion:

- Translational

- Vibrational

- Roational

Translational motion in QM is described by the particle in a box (PIB - Chap 4). Vibrational motion is described by the harmonic oscillator (sec 7.1). Rotational motion is described by a particle on a ring (2D - Sec 7.2) or sphere (3D - Sec 7.3).

It is convenient for us to discuss all of these modes of motion/energy as separate things, but in the end, we cannot separate the motions from each other and so the final "solution" is a bit messy.

I can draw an analogy to understanding 3D space by using the dimensions of x, y, and z. I could explain the concept of an x-axis (1D), or position along the x-axis and then tell you that a y-axis and z-axis follow all of the same concepts as the x-axis. We then put them all together in order to describe 3D space. In this case, x, y, and z fully describe all 3D space and we need to know nothing more than the x, y, and z values to know where an object is in this 3D space; we sometimes refer to x, y, and z as a "complete set."

As you have already learned from the PIB example, all quantum mechanical models have an operators, eigenfunctions, and eigenvalues. So i hope you have your "Hamiltonian Table" i passed out in class. I will link it here shortly. At this point you should have the following information in the table:

- Translational (1D)

- Translational (3D)

- Vibrational

...and by the end of this Chapter the entire table should be complete.

Vibrational Motion

Points of noteworthiness

- The Schrodinger Equation now includes a potential of 1/2kx^2

- The wavefunctions are related to the Hermite polynomials (eq. 7.5)

- The energies are fairly straightforward (eq. 7.6)

-...and this can all be pictured as shown in Fig 7.2 and 7.3 which you have already made in Mathematica.

To have a slightly different (physics) approach to this problem, i suggest you watch the following videos:

Quantum Harmonic Oscillator Part 1 (5:45 mins)

Quantum Harmonic Oscillator Part 2 (7:38 mins)

he goes into LOTS of detail about solving...the answer is Hermite polynomials!